Building upon the idea of representation, we will
discuss how images are represented in digital form. We'll work up to
it, first starting with how color is represented (which is based on
the physiology of the human eye), then looking at images as
rectangular arrangements of spots of pure color. Finally, we'll
calculate the file size of an image and discuss one way of
compressing the file so that it is smaller and therefore
faster to download. This compression is, in fact, a different
representation of the information.

Standard Colors

In the present day, modern browsers support 140 color names. This means
that we can use color names such as black, aqua,
or chocolate as values for the CSS properties that except
a color value, such as color, background-color, etc.
Many years ago, browsers could only support 17 color names, known as standard
colors: aqua, black, blue, fuchsia, gray, green, lime, maroon, navy, olive,
orange, purple, red, silver, teal, white, and yellow. However, later that
list was expanded with 123 more colors. W3Schools maintains
a complete list of the
140 recognized color names. While you can achieve a lot by using
only these named colors, very often you want something more specific from the color
spectrum. It turns out, we can use numerical codes to refer to colors, because
inside the computer, colors are represented by numbers, in the same
way as text characters (ASCII and unicode). How? For that, we need to understand
additive colors and color vision.

Additive (RGB) Colors and Color Vision

Our retinas happen to have rod-shaped cells that are sensitive to
all light, and cone-shaped cells that come in three kinds:
red-sensitive, green-sensitive, and blue-sensitive. Therefore, there
are three (additive) primary colors: Red, Green and Blue or RGB. All
visible colors are seen by exciting these three types of cells in
various degrees. (For more information, consult these Wikipedia articles
on additive color
and color
vision.)

Color monitors and TV sets use RGB to display all their colors,
including yellow, chartreuse, you name it. So, every color is some amount
of Red, some amount of Green, and some amount of Blue.

On computers, RGB color components are standardly defined on a scale from 0 to 255,
which is 8 bits or 1 byte.

The first two ways are self-explanatory, since they use decimal numbers
and percentage values with which you are familiar. In the following,
we will explain the meaning of the hexadecimal color codes such as
#40E0D0. The # sign is used in this case
to simply indicate that the sequence of digits and letters that is
following it should be represented as a hexadecimal code.

Hexadecimal

People use decimal (base 10), computers use binary (base 2), but
programmers often use hexadecimal (base
16) for convenience.

Binary numerals get long very fast. It is not easy to remember 24
binary digits, but you can more easily remember 6 hexadecimal
digits. Each hexadecimal digit represents exactly four binary digits
(bits). (This is because 24=16.)

One way to understand hexadecimal is by analogy with decimal, but
we're all so familiar with decimal numerals that our reflexes get in
the way. (In fact, humans throughout history have used many different
numeral
systems; decimal is not sacrosanct.) So, we first need to break
down decimal notation so that you can see the analogy with
hexadecimal. For now, we'll stick with two-digit numerals, but the
same ideas extend to any larger numbers.

Decimal notation works by organizing things into groups of ten,
then counting the groups and the leftovers: Suppose you had a bunch of
sticks on the ground and you bundled them all into groups of 10 with
some left over (fewer than 10). Now, use a symbol to denote the number
of bundles and another symbol to denote the number of sticks left
over. You've just invented two-digit numbers in base 10.

Hexadecimal: Do the same thing with bundles of 16, and you've invented
two-digit numbers in base 16. For example, if you had thirty-five sticks
, they could be bundled into two
groups of sixteen and three left over, so the hexadecimal notation is 23.
Careful! That numeral isn't the decimal number twenty-three! It's still
thirty-five sticks, but we write it down in hexadecimal as 23.

To distinguish a decimal numeral from a hexadecimal numeral, we use
subscripts. So, to say that thirty-five sticks is written 23 in
hexadecimal, we can write:

3510 = 2316

Both decimal and hexadecimal notations are based on position. We say that 2316 means 3510 because
it's a "2" in the sixteens position and "3" in the ones
position, just like 3510 has a "3" in the tens position
and a "5" in the ones position.

Let's take another example. Suppose we have 2610
sticks. That's one group of 16 and 10 left over. How do we write
that number in hexadecimal? Is it 11016? That is, a "1"
in the sixteens place followed by a "10" in the ones
place? No; that would be confusing, since it would look like a
three-digit numeral. We need a symbol that means ten. We can't use
"10," since that's not a single symbol. Instead, we use "A"; that is,
A16=1010. Similarly, "B" means 11, "C" means
12, "D" means 13, "E" means 14, and "F" means 15. We don't need any
more symbols, because we can't have 16 things left over, since that
would make another group of 16. The following table summarizes these
correspondences and what we've done so far.

The correspondence between decimal numerals and hexadecimal numerals

Decimal

0

1

...

9

10

11

12

13

14

15

16

17

18

...

28

29

30

31

32

33

34

35

36

Hexadecimal

0

1

...

9

A

B

C

D

E

F

10

11

12

...

1C

1D

1E

1F

20

21

22

23

24

To convert a big decimal number to hexadecimal, just divide. For
example, 23010 divided by 16 is 1410 with a
remainder of 610. Thus, the hexadecimal numeral is
E616. To convert a hexadecimal number to decimal, just
multiply: E616=E*16 + 6 = 14*16 + 6 = 230.

Exercise 1

Try the following conversions as an in-class exercise. You can use
a calculator, you can ask your neighbors, anything you like.

Dec

Hex

Dec

Hex

7

22

26

100

127

149

240

255

You can check your work with the following form:

Decimal

Hexadecimal

Converting Hexadecimal to or from Binary

Now that we know both hexadecimal and binary, you can convert binary to
hexadecimal (and vice versa). However, you would probably do so by
converting the binary number to decimal and then the decimal number to
hexadecimal. There's a better way, involving almost no arithmetic (or,
rather, all the arithmetic is with one-digit numbers you can add in your
head). Indeed, this technique is the reason that computer
scientists like using hexadecimal.

Example 1

Let's start with an example. Suppose you need to convert the following
from binary to hexadecimal:

01010100 = ??16

What we're going to do is to take the bits in chunks of four bits, so
to mark the chunks we'll insert a period in the middle of the number:

0101.0100 = ??16

Now, we just convert each chunk directly into hex. The first chunk,
0101, is just the number 5. The second chunk, 0100, is just the number
4. Those are already in hex, so we are done:

0101.0100 = 5416

(Try doing it via decimal, to check. The decimal value corresponding
to both of these is 80+4=84.)

Example 2

Let's do another one, this time with slightly larger values:

10101100 = ??16

Again, take the bits in chunks of four bits:

1010.1100 = ??16

Now, we just convert each chunk directly into hex. The first chunk,
1010, is 8+2 or 1010, which is the digit A in hex. The second
chunk, 1100, is 8+4 or 1210, which is the digit C in hex. So
we are now done:

1010.1100 = AC16

(Again, check our work by doing it via decimal. The decimal value
corresponding to both of these is 160+12=172.)

Explanation

Why does this work? Suppose we needed to convert 172 from decimal to
hex: our first step would be to divide the number by 16. In binary,
moving the binary point to the left by one place is equivalent to
dividing by two, so moving the binary point four places is equivalent to
dividing by 16. So when we put a period in the middle of the 8-bit
binary number, it is exactly the same as dividing by 16. We then have
the quotient to the left of the binary point, and the remainder to the
right of the binary point. Just convert each to hex, and we are done.

Notice that the only arithmetic we have to do is converting each chunk
of four bits to the equivalent hex digit. The mental arithmetic
involved is limited: we know that (1) we are adding one-digit numbers,
(2) at most four of them, and (3) the sum will always be less than 16.

Colors using Hexadecimal

We already know that every color in a computer is a combination
of some amount of each of the three primary colors: red, green and blue.
The amounts are always given in the same order: red, green,
blue. The amounts are numbers from 0 to 25510, or, in
hexadecimal, 00 to FF16. Each primary is expressed as a
two-digit numeral in hexadecimal, using a leading zero if necessary so
that the numeral is always two digits. Three pairs of hexadecimal
digits completely specifies a color. Finally, the notation for a
color always starts with a pound sign (#). For example, a color like
(35, 230, 10) would be written #23E60A.

Exercise 2

Experiment with defining a color numerically. In the form below, enter
a color value in the syntax #RRGGBB and press return/enter. The box will
change its background color to display the entered color value.

Image Representation

Now that we know how to represent a color, we can represent
images. You can think of an image as a rectangular 2D grid
of spots of pure color, each represented as RRGGBB. A spot of pure
color is called a pixel, short for
picture element, the atom of a picture. Pixels are better seen if you
blow up an image several times; here are some examples.

Every image on the computer monitor is represented with pixels,
including the windows themselves! Such images are saved in files that,
in addition to the image data, contain information on the size of the
image, the set of colors used, the origin of the image, etc. Depending
on how exactly this information is saved, we refer to them as image
formats GIF, JPG, PNG, and BMP are some of the well-known
image formats. We will talk more about image formats below. For
now, we will focus on the number of pixels and the representation of
each pixel, and consequently, the file size of the image.

We said above that the amount of each primary color is a number
from 0 to 25510 or 00 to FF16. It is no
coincidence that this is exactly one byte (8 bits). A byte is a
convenient chunk of computer memory, so one byte was devoted to
representing the amount of a single primary color. Thus, it takes 3
bytes (24 bits) to represent a single spot of pure color.

With 256 values for each primary,
that yields 256 x 256 x 256 = 16,777,216 colors. Humans can distinguish
over 10 million colors, so 24-bit color is sufficient to represent more
colors than humans can distinguish. All modern monitors use this
so-called 24-bit color. Some old monitors used 16-bit or 8-bit color,
which were relatively impoverished, being only able to represent 65,536
colors (for a 16-bit monitor) or 256 colors (for an 8-bit monitor). Of
course, a black-and-white monitor can only represent two colors, which
could be called 1-bit color.

Image size and download time

Every pixel needs 24 bits or 3 bytes to be stored. Let's suppose you are going
to take pictures of all your 30 class peers for a class website, using your
iPhone4 camera. According to the
phone specifications,
its screen has 2592 x 1936 pixels, which amounts to about 5 million pixels, or 5MP.
(The size of one pixel is 1.75 µm.) Thus, if every pixel takes 3 bytes, and a photo
with your camera has 5MP, to store the image you need 15MB (mega bytes). For all
your peer photos, you will need 30 x 15MB = 450 MB.

Imagine now that you put all these photos online on your website, in one
single page (using the attributes width and height to make them fit in one screen), and
then you send the link to this page to your parents. They might have an average
Internet connection (e.g. Verizon offers offers a 1-3 Mbs (mega bit per second)
to non-FiOS subscribers).

The amount of time that it will take to load a page with all these pictures on
your parent's computer can be calculated as below:

content size (450MB) x 8 bits/byte / 1Mbs = 3600 seconds or 1 hour.

If each of your photos would have been around 100KB (as we asked you in HW2),
then the amount of time to load all of them on the page would have been 24 seconds.

So, how do we get our images to be so small in size? There are two ways: resizing
(decrease the amount of pixels for the image) and you have already seen this in lab,
and compressing (decreasing the amount of bytes per pixel) and we will discuss
this in the next section.

Compression

Short of making our images smaller (fewer pixels), what can we do
to speed up the downloads? We can compress the files.

There are two classes of compression techniques:

lossless compression, where clever encoding allows the number of
bytes to be reduced but where the original image can be perfectly
reconstructed from the compressed form, and

lossy compression, where we discard less-important information
in order to reduce the amount of information to be stored or
transmitted.

We will look in detail at one kind of lossless compression, which
is indexed color (GIF encoding), because it gives us a window into the
kinds of ideas and techniques that matter in designing
representations of information.

Indexed Color

The idea behind indexed color is that if a particular color is used
many times in an image, we can create a "shorthand" for it. In fact,
if we limit the number of colors, each one can be assigned a
shorthand. What will be confusing is that the colors are, of course,
represented as numerals and so are the shorthands! For example,
instead of saying (for the umpteenth time), color #D619E0, we'll just
say, for example, color number 5. This will only work, however, if the
shorthands really are shorter. They are, and we'll see exactly how
much.

One way to think about indexed color is that we are
creating a "paint-by-numbers"
picture. We choose:

the numbered list of colors

what color (number) each pixel is

Example: Two-color image

Imagine that a 300x500 picture uses only two colors,
say red and yellow. Suppose we make up a table of colors (two entries)
and then represent the image with an array of "color indexes," like a
paint-by-numbers set.

What is the numbered list of colors? There are just two:

index

color

0

#FF0000

1

#FFFF00

We then paint the picture using just two numbers, 0 and 1. A
zero means a pixel is red, and a one means the pixel is yellow.

How many bits does it take to represent this image? Well, there
are 300x500 or 150,000 pixels, but each one is just 1 bit, so it
takes 150,000 bits or 150,000/8 = 18,750 bytes or about 18 kB.
Compare that with the 450 kB (300 x 500 x 3 byte/pixel) of the
normal representation, and you can see
this is much smaller. In fact, it's 1/24th the
size, since each pixel takes 1 bit to represent rather than 24.
It'll be 24 times faster to download.

What about that table of colors? That's called the color
palette, by analogy with an artist's palette. That has to
be represented too. Otherwise, the browser would know there were
only two colors in the picture, but wouldn't know what colors they
are. There are two entries in this palette, each of which is 3 bytes
(24 bits), so add at least 6 more bytes to the representation.

You can see the general scheme at work: we create a table of all
the colors used in the picture. The shorthand for a color is simply
its index in the table. We will limit the table so that the
shorthands will be at most 8 bits. Since the shorthands are all
replacing 24-bit color specifications, the shorthand is at
most one-third the size. In the example above, the shorthand is
1/24th the size.

Example: Four-color image

Let's continue with the example. What is the
file size if the image uses 4 colors, say red, yellow, blue and lime?
In that case, the table looks like this:

index

color

00

#FF0000

01

#FFFF00

10

#0000FF

11

#00FF00

As you can see, the shorthand is now two bits instead of one.
Therefore, the 150,000 pixels require 300,000 bits or
300,000/8=37,500 bytes or about 37.5kB. Obviously, this is about
twice the size of the previous example, since each shorthand is now
twice as big. Nevertheless, it's still much smaller than the 450 kB
uncompressed file.

What about the size of the palette? That's now twice as big,
too. Four entries at 3 bytes each adds 12 bytes to the file size,
which is a negligible increase to the 37.5 kB.

What's the pattern here? The number of colors in the original
image determines the size of the palette, which determines the number
of bits in each shorthand, which then determines the size of the file
as a whole. The shorthand for a color is simply the binary numeral
for the row that the color is in the table. For example, the color
red in the last example was in row zero (00 in binary) and the color
lime was in row 3 (11 in binary).

You can see that the number of bits required for each pixel is the
key quantity. This quantity is called bits per pixel or "bpp." It's
also often called "bit depth" so that the file size of an image is
just width x height x bit depth, almost as if it were a
physically 3D box.

Finally, we can state the rule:

The bit depth of an image must be large enough so that the
number of rows in the table is enough for all the colors. If the bit
depth is d, the number of rows in the table is 2d.

Here's the exact relationship, along with the size of a 300x500 image:

Mapping bit-depth to number of colors

bit-depth

max colors

file size of 300x500 image

1

2

18kB

2

4

37kB

3

8

55kB

4

16

73kB

5

32

91kB

6

64

110kB

7

128

128kB

8

256

147kB

Exersise 3

Consider an image that is 80 x 100 (pixels).

How many bytes are needed to represent this image if it's black and
white? Don't forget to represent the color table.

How many bytes if the image uses 4 colors?

How many bytes if the image uses 16 colors?

How many bytes if the image uses 17 colors?

In summary, you can reduce your image file size by using fewer
colors. Of course, this may reduce the quality of your image. It's a
tradeoff.

We will continue to discuss file size calculations in Lab 5 and Assignment 5.